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Chapter 0: Problem 28

Find an equation of the line: with \(m=7,\) containing \((1,7).\)

Short Answer

Expert verified
The equation of the line is \( y = 7x \).

Step by step solution

01

Identify the given information

The slope of the line is given as \(m = 7\), and the line passes through the point \( (1, 7) \).
02

Use the point-slope form of the equation

The point-slope form of the equation of a line is \[ y - y_1 = m(x - x_1) \]. Substitute \(m = 7\), \(x_1 = 1\), and \(y_1 = 7\) into the formula.
03

Substitute the values into the point-slope form

Plug in the values to get the equation: \[ y - 7 = 7(x - 1) \].
04

Simplify the equation

Distribute the slope and simplify: \[ y - 7 = 7x - 7 \]. Then, add 7 to both sides: \[ y = 7x \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Point-Slope Form
In the context of a line equation, the point-slope form is essential. It allows us to write the equation given a point on the line and the slope.
The point-slope form is expressed as: \( y - y_1 = m(x - x_1) \).
Here, \(m\) is the slope, and \( (x_1, y_1) \) is a point on the line.
This form is useful because it directly incorporates both the slope and a known point. This makes it straightforward to generate the line's equation.
Employing this form, we substitute the known values into our formula, leading to easier transformations.
Slope
The slope of a line is a measure of its steepness and direction. It is often denoted by \(m\).
Mathematically, the slope is the ratio of the vertical change to the horizontal change between any two points on the line. This can be written as: \[ m = \frac{\text{rise}}{\text{run}} = \frac{\text{change in } y}{\text{change in } x} \]
A positive slope indicates an upward trend from left to right, while a negative slope indicates a downward trend.
Given in the problem, \(m = 7\) suggests a steep upward line. Understanding this concept is crucial for accurately determining the equation of a line.
Linear Equations
A linear equation represents a straight line on the coordinate plane. It can be expressed in various forms such as point-slope, slope-intercept, and standard form.
The general form of a linear equation is: \[ y = mx + b \]
Here, every term corresponds to specific attributes of the line:
  • 'y' and 'x' are coordinates of any point on the line.
  • 'm' is the slope.
  • 'b' is the y-intercept, the point where the line crosses the y-axis.

Converting our derived point-slope equation into this form helps in understanding and using the line equation in various contexts.
Simplification
Simplification is the process of making an equation or expression simpler. In our exercise, it involves distributing and combining like terms.
Starting from \( y - 7 = 7(x - 1) \) and distributing the 7 to obtain: \[ y - 7 = 7x - 7 \]
Next, adding 7 to both sides cleans the equation up: \[ y = 7x \]
This simplified form is much easier to work with.
Simplification is a critical step in making equations usable and easily understandable.

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Most popular questions from this chapter

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